Answer:
a. Convergence of a sequence is a fundamental concept in mathematics that describes the behavior of a sequence as its terms approach a specific value. In particular, a sequence {sn}1∞ converges to s if, as the index n increases without bound, the terms of the sequence get arbitrarily close to the value s. This can be mathematically expressed as follows:
For any positive real number ε, there exists a positive integer N such that for all n > N, |sn - s| < ε.
In simpler terms, this definition states that no matter how small we choose a positive number ε to be, we can always find a point in the sequence beyond which all subsequent terms are within ε distance from the limit s.
To illustrate this concept visually, imagine a number line with dots representing each term of the sequence. As we progress along the number line by increasing the index n, the dots gradually get closer and closer to each other. Eventually, they converge to a single point at s, indicating that the sequence converges to s as n approaches infinity.
It is important to note that not all sequences converge. Some sequences may exhibit different behaviors such as divergence or oscillation. A divergent sequence does not have a specific limit and its terms do not approach any particular value as n increases. On the other hand, an oscillating sequence alternates between multiple values without converging to a single limit.
Convergence of sequences has numerous applications in various branches of mathematics and other fields such as physics and engineering. It provides a foundation for understanding limits, continuity, and convergence of series, among other concepts.
Overall, convergence of a sequence refers to the behavior of its terms as they approach a specific value when the index increases without bound. It is a fundamental concept in mathematics with wide-ranging applications.
Explanation: